If the character table of a group G has a row (corresponding to an irreducible character) with precisely two nonzero entries, then G has a unique minimal normal subgroup N which is necessarily an elementary abelian p-grovφ for some prime p.

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Sequences with optimal correlation properties are much sought after for applications in communication systems. In 1980, Alltop (IEEE Trans. Inf. Theory 26(3):350-354, 1980) described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation. Subsequently these sequences were used to construct mutually unbiased bases (MUBs), a structure of importance in quantum information theory. The key feature of this cubic function is that its difference function is a planar function. Functions with planar difference functions have been called Alltop functions. This paper provides a new family of Alltop functions and establishes the use of Alltop functions for construction of sequence sets and MUBs.

In this short note, we use compositions to study the partition perimeter, a statistic defined to be one less than the sum of the number of parts and the largest part. This leads to some generalizations of known theorems. Our main result is a combinatorial proof that for m > 2 and n > m, there are strictly more m-distinct partitions than m-regular partitions with perimeter n, which provides an affirmative answer to a question from a recent paper of Amdeberhan et al. Additional refinements and applications of this are still being investigated.

We study the situation of a finite group